3. MASS-TO-LIGHT RATIOS AND `MAXIMUM DISK'

The distribution of the products
(M / L)1/2
(z /
R)-1
in the
Kregel et
al. (2005)
sample is shown in fig. 2. Thirteen of the
fifteen disks have 1.8
(M / L)1/2
(z /
R)-1 3.3. The values
of the outliers may have been overestimated (see
Kregel et
al. 2005).
Excluding these, the
average is < (M / L)1/2
(z /
R)-1
> = 2.5 ± 0.2 with a
1 scatter of 0.6.
The near constancy of the product can be used with M / L
based on stellar population synthesis models to estimate the axis
ratio of the velocity ellipsoid. Conversely, the upper scale of
Fig. 2 indicates that a typical (I-band)
M / L of a galactic stellar disk is of order unity and
varies for the majority systems between 0.5 and 2.

Figure 2. Histogram of the product
(M / L)1/2
(z /
R)-1
from stellar kinematics in edge-on galaxies. Except for two outliers
the distribution of (M / L)1/2
(z /
R)-1
is rather narrow. The outliers are ESO 487-G02 and
564-G27; data for these galaxies are less complete
than for the other
ones. Along the top we show the values of M / L implied by
z /
R
= 0.6. (From
Kregel et
al. 2005)

It is possible to relate the axis ratio of the velocity ellipsoid to the
flattening of the stellar disk h / hz
(van der Kruit
& de Grijs 1999).
In the
radial direction the velocity dispersion is related to the epicyclic
frequency through the Toomre parameter Q for local stability.
The Tully-Fisher relation then relates
this to the integrated magnitude and hence to the disk scalelength.
In the vertical direction the scaleheight and
the velocity dispersion relate through hydrostatic equilibrium.

The observed constancy of (M / L)1/2
(z /
R)-1
implies that the flattening of the disk h / hz
is proportional
to Q(M / L)1/2.

The mass-to-light ratio is a crucial measure of the contribution of the
disk to the rotation curve and the relative importance of disk and dark
halo mass in a galaxy. In the `maximum disk hypothesis' the disk
contribution is optimized such that the amplitude of the disk-alone
rotation curve is as large as the observations allow. In a maximal disk,
the ratio between the disk-alone rotation curve and the observed one
will be a bit lower than unity to allow a bulge contribution and let
dark halos have a low density core. A working definition is
Vdisk / Vrot = 0.85 ± 0.10
(Sackett 1997).

For an exponential disk, the ratio of the peak rotation velocity of
the disk to the maximum rotation velocity of the galaxy
(Vdisk / Vrot) is

(13)

Using eqn. (5) and eqn. (8) this can be rewritten as

(14)

So we can estimate the disk contribution to the rotation curve from
a statistical value for the flattening (see also
Bottema 1993,
1997,
van der Kruit
2002).
For the sample of
Kregel et
al. (2002)
this then results in Vdisk / Vrot =
0.57 ± 0.22 (rms scatter). In the dynamical analysis of
Kregel et
al. (2005),
the ratio Vdisk / Vrot
is known up to a factor
z /
R and
distance-independent. For
z /
R = 0.6,
vdisk / vrot = 0.53 ± 0.04,
with a 1 scatter of
0.15. Both estimates agree well. Thus, at
least for this sample, the average spiral has a submaximal disk.

The values obtained for individual galaxies are illustrated in
figs. 3 and 4.
Most galaxies are not `maximum-disk'. The
ones that may be maximum disk have a high surface density
according to fig. 2. From the panels we also note
that disk that are maximal appear to have more anisotropic velocity
distributions or are less stable according to Toomre Q,

Figure 3. The contribution of the disk to
the amplitude of the rotation curve Vdisk /
Vrot. for a sample of 15 edge-on galaxies as a function
of the rotation velocity itself. The horizontal dashed lines are the limits
of 0.85 ± 0.10 from
Sackett (1997),
which would indicate maximal disks. The axis ratio of the velocity
ellipsoid is assumed to be 0.6. The grey lines correspond to collapse
models of
Dalcanton et
al. (1997).
The two without error bars are the same
galaxies as the outliers in fig. 2.
(From
Kregel et
al. 2005)

Figure 4. Stellar dynamics parameters for
edge-on galaxies. (a) The axis ratio of the velocity ellipsoid as
a function of Vdisk / Vrot for
Q = 2.0. (b) Vdisk / Vrot as
a function of Q for an assumed axis ratio of
the velocity ellipsoid of 0.6. (From
Kregel et
al. 2005)